Discussion
Consider the following statements.
I. Every compact Hausdorff space is normal.
II. Every metric space is normal.
Which one of the following is correct?
Answer: A
Both I and II are TRUE.
I: in a compact Hausdorff space, a point and a disjoint compact (closed) set can be separated, and repeating the argument with finite subcovers separates two disjoint closed sets. So the space is normal.
II: in a metric space, disjoint closed sets \(A\) and \(B\) are separated by the open sets \(\{x : d(x, A) \lt d(x, B)\}\) and \(\{x : d(x, B) \lt d(x, A)\}\). So every metric space is normal. Options declaring either statement false are wrong.
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